Dynamical model and nonextensive statistical mechanics of a market index on large time windows

The shape and tails of partial distribution functions (PDF) for a financial signal, i.e., the S&P500 and the turbulent nature of the markets are linked through a model encompassing Tsallis nonextensive statistics and leading to evolution equations of the Langevin and Fokker-Planck type. A model originally proposed to describe the intermittent behavior of turbulent flows describes the behavior of normalized log returns for such a financial market index, for small and large time windows, and both for small and large log returns. These turbulent market volatility (of normalized log returns) distributions can be sufficiently well fitted with a ${\chi }^2$ distribution. The transition between the small time scale model of nonextensive, intermittent process, and the large scale Gaussian extensive homogeneous fluctuation picture is found to be at ca. a 200 day time lag. The intermittency exponent $\kappa$ in the framework of the Kolmogorov log-normal model is found to be related to the scaling exponent of the PDF moments, thereby giving weight to the model. The large value of $\kappa$ points to a large number of cascades in the turbulent process. The first Kramers-Moyal coefficient in the Fokker-Planck equation is almost equal to zero, indicating “no restoring force.” A comparison is made between normalized log returns and mere price increments.